1. Non Leap YearLeap Year DateDay NumberMod 7Day NumberMod 7 13-Jan136 6 13-Feb442 2 13-Mar722733 13-Apr10351046 13-May13301341 13-Jun16431654 13-Jul19451956 13-Aug22512262 13-Sep25642575 13-Oct28662870 13-Nov31723183 13-Dec34643475 Friday the 13th To see in which months Friday the 13 th can occur in the same calendar year, it’s helpful to number the dates 1 – 365. In the table below, the 13 th of every month is numbered as it falls in the year and each 13 th has its equivalent mod 7 day number: From the table, it’s evident that in a non-leap year, February and March 13 th are both equivalent to 2 mod 7, so a Friday the 13 th in February will be followed by a Friday the 13 th in March. These are the only months that have consecutive equivalent values for the 13 th in mod 7. In order for the 13 th of February and March to occur on a Friday, January 13th must be on a Tuesday (6 mod 7 is on a Tuesday, 0 mod 7 is on Wednesday, 1 mod 7 is on a Thursday and 2 mod 7 is on a Friday). This would indicate that January 1 st would need to fall on a Thursday. From this table, we can see that there must be a Friday the 13 th every year, because the 13 th of the months fall on some value 0 – 6 at some point in the calendar year – this means that the 13 th will fall on every day of the week at some point in the calendar, and this must include Friday. The most Friday the 13ths that can occur in one year would occur in February, March and November in a non-leap year. In a leap year, three Friday the 13ths could occur in the months January, April and July.

2. To forecast the years where January 1 st will fall on a Thursday, we can look at recent years and find a pattern: Number of Days in the Following Year # Days in Followi ng Year mod 7 Sum of Days Mod 7 Since The Last Year that Started on Thursday 2004Thursday36622 2005Saturday36513 2006Sunday36514 2007Monday36515 2008Tuesday36620 2009Thursday36511 2010Friday36512 2011Saturday36513 2012Sunday36625 2013Tuesday36516 2014Wednesday36510 2015Thursday36511 2016Friday36623 2017Sunday36514 2018Monday36515 2019Tuesday36516 2020Wednesday36621 2021Friday36512 2022Saturday36513 2023Sunday36514 2024Monday36626 2025Wednesday36510 2026Thursday36511 2027Friday36512 2028Saturday36624 2029Monday36515 2030Tuesday36516 2031Wednesday36510 2032Thursday36622 2033Saturday36513 2034Sunday36514 2035Monday36515 2036 36620 2037Thursday3651

3. By using the sum of days in each year mod 7, we can predict the years that will have consecutive months with Friday the 13 th (in February and March, i.e. when January 1 st is on Thursday): 2009, 2015, 2026, 2032, 2037, 2043, 2054, 2060, 2065, 2071, 2082, 2088, 2093, 2099.